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Integrating for Fourier Series

  1. Jun 1, 2010 #1
    1. The problem statement, all variables and given/known data
    For positive integers m and n, calculate the two integrals:

    [tex]\frac{1}{L}\int^{L}_{-L}sin(\frac{n \pi x}{L})sin(\frac{m \pi x}{L})dx[/tex] and [tex]\frac{1}{L}\int^{L}_{-L}sin(\frac{n \pi x}{L})cos(\frac{m \pi x}{L})dx[/tex]


    2. Relevant equations
    [tex]\int u v' dx = [u v] - \int u' v dx[/tex]


    3. The attempt at a solution
    For the first one, I can't seem to get anything other than 0 for the integral (but not in the normal way). If I work through, I end with [tex]I = \frac{n^{2}}{m^{2}}I[/tex] and that makes no sense at all. Every other part of the integral I find cancels to 0 as they all include [tex]sin(\frac{n \pi x}{L}) or sin(\frac{m \pi x}{L})[/tex] which will be 0 as n and m are integers. What am I doing wrong?

    Edit: I plugged the two integrals into Mathematica and found the second one to be 0, which I calculated, but the first one is not. What do I need to change in my working?
     
    Last edited: Jun 1, 2010
  2. jcsd
  3. Jun 2, 2010 #2
    You could use the relations

    [tex] sinAsinB=\frac{cos(A-B)-cos(A+B)}{2} [/tex]

    [tex] sinAcosB=\frac{sin(A+B)+sin(A-B)}{2} [/tex]

    This will make your integration easy.
    It is better to view graphically too. Try to plot multiplication of two sine waves of different frequencies ( a sin and cos wave of different frequencies also) and see graphically. You can guess their average value graphically. In fact that is the result of your integration also.
     
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